Green functions with the use of the stochastic integral

Abstract

The Euler–Maruyama approximation to the discrete-time Langevin equation with multiplicative noise is used to construct a functional representation for the generating functional of correlation and response functions. Closed loops of propagators are shown to be absent in the corresponding perturbation theory, which reproduces the results of averaging the iteration solution of the stochastic difference equation. The continuous-time generating functional is defined in terms of perturbation theory by the continuous-time limit of the ordinary integral sums of the Feynman diagrams of the discrete-time perturbation theory. Compensation terms needed to exclude graphs with self-contracted propagators due to the usual Feynman rules in the continuum case are described in terms of the normal-form representation of perturbative field theory. The generating functional for the Stratonovich interpretation is obtained as the white-noise limit of multiplicative Ornstein–Uhlenbeck process noise.